Locally solid topological lattice-ordered groups

TL;DR

This paper initiates a systematic study of locally solid topological lattice-ordered groups, characterizing their structure and properties, and exploring their homomorphisms, boundedness, and completions.

Contribution

It provides the first comprehensive characterization and analysis of locally solid topological lattice-ordered groups, including their generation by lattice pseudometrics and properties of homomorphisms.

Findings

Characterization of locally solid topological lattice-ordered groups via lattice pseudometrics

Analysis of lattice group homomorphisms in these groups

Relationship between order-bounded and topologically bounded subsets

Abstract

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Robert-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is locally solid if and only if it is generated by a family of translation-invariant lattice pseudometrics. We also investigate (1) the basic properties of lattice group homomorphism on locally solid topological lattice-ordered groups; (2) the relationship between order-bounded subsets and topologically bounded subsets in locally solid topological lattice-ordered groups; (3) the Hausdorff completion of locally solid topological lattice-ordered groups.